Syllogism Shortcuts with Examples - Lesson 2

In our Syllogism Shortcut Lesson 1, we have discussed about the Introduction of Syllogism. So before going to read this Lesson 2, it is advisable to read that Lesson 1 here. In this post we shall discuss some important rules with Examples.

Important Rules for Syllogism :

1. Every deduction should contain three and only three terms.
2. The middle term must be distributed at least once in the premises.
3. If one premise is negative the conclusion must be negative.
4. If one premise is particular the conclusion must be particular. 
5. If both premises are negative no conclusion can be drawn.
6. If both premises are particular no conclusion can be drawn.
7. No term can be distributed in the conclusion if it is not distributed in the premises.

Now we shall discuss these rules with examples.

Syllogism Examples :

Example 1 : 

All dogs are cats ---- (1)

All cats are pigs ----- (2)

As the first statement is a universal affirmative statement, the subject (dogs) has to be distributed (✔) and the predicate (cats) not distributed (☓). As the second statement is also universal affirmative the subject cat is distributed (✔) and the predicate pigs not distributed (☓).

The above is arrived at on the basis of Table II.

The middle term ("cats" is the middle term as it occurs in both the premises) is distributed once in the premises. Hence it satisfies Rule [2]. As "Dogs" is distributed in the premise and "Pigs" undistributed, in the deduction also, they should appear accordingly. The type of statement that satisfies both of them is universal affirmative statement, i.e., a statement with "All". Hence the answer will be

All dogs are pigs

The answer cannot be "All pigs are dogs" because Rule [7] states that no term can be distributed in the conclusion if it is not distributed in the premises. As "pigs" is not distributed in premise it cannot be distributed in the conclusion (because if we take "All pigs are dogs", then the subject "pigs" will be distributed). Hence the conclusion "All pigs are dogs" is wrong.

Example II :

All  cats are dogs ---- (1)

All cats are pigs ----- (2)

Statement I is Universal affirmative and hence the subject "cats" is distributed and the predicate "dogs" is not distributed as per Table II.

Statement II is also Universal affirmative and hence the subject "cats" is distributed and the predicate "pigs" is not distributed as per Table II.

here the middle term "cats" ("Cats" is the middle term as it is occuring in both the premises) is distributed; hence we can draw a conclusion.

The answer should contain the terms "dogs" and "pigs" and both the terms are not distributed. Referring to Table II, we find that this is possible only in Particular Affirmative [the conclusion cannot start with the qualifier "All" as the subject in "All" should be distributed]. According to Rule 7 a term cannot be distributed in the conclusion if it is not distributed in the premises. So the answer will be

"Some dogs are Pigs" or "Some Pigs are Dogs"

Example III :

All Dogs are Cats ------ (1)

All Pigs are Cats ------- (2)  

Statement (1) is universal affirmative and hence the subject "Dogs" is distributed and the predicate "Cats" is not distributed. In statement (2) which is also a universal affirmative, the subject "Pigs" is distributed and the predicate "Cats" is not distributed. This is arrived at on the basis of Table II.

The middle term "Cats" ["cats" is the middle term as it occurs in both the statements] is not distributed in either one of the two statements. From Rule [2], which states that the middle term should be distributed at-least once in the premises for drawing a conclusion, we cannot draw any conclusion in this case.

Example IV :

All Cats are Dogs ------ (1)

Some Cats are Pigs ------- (2)  

The first statement is a universal affirmative premise and hence the subject "cats" is distributed (✔) and predicate "dogs" is undistributed (☓). The second statement is particular affirmative and hence both the subject "cats" and the predicate "pigs" are undistributed (☓) as per Table II. As we have a particular premise, the conclusion should also be a particular one as per Rule [4]. The middle term is distributed hence  we can draw a conclusion. So the answer will be

Some Dogs are Pigs or Some Pigs are Dogs

Example V :

All Dogs are Cats ------ (1)

No Cats are Pigs ------- (2) 

As the first premise is universal affirmative the subject (dogs) is distributed and the predicated (cats) is undistributed. In the second premise which is universal negative the first term (cats) and the second term (pigs) are both distributed (as per Table II). As the middle term is distributed at-least once in the premises, Rule [2] is satisfied and hence we can draw a conclusion.



From Rule [3] which states that if one of the premise is negative the conclusion should be negative, the answer should be a negative one and as both the terms dogs and pigs are distributed the conclusion should be a universal negative statement. Hence the answer will be



No dogs are pigs or

No pigs are dogs

Example VI :

All dogs are Cats ---------- (1)

Some cats are not pigs --------- (2)

Since the first statement is universal affirmative, "dogs" is distributed and "cats" is not distributed. Since the second statement is particular affirmative, "cats" is not distributed and "pigs" is also not distributed (as per Table II).

In the above given example no conclusion can be drawn as Rule [2] which states that the middle term ("Cats" in the example above as it occurs in both the premises) should be distributed at least on in the premises not satisfied.